This article will briefly cover: theory and applications of the magnetorotational instability in astro- and geophysics, highlighting its historical role in understanding the onset of turbulence in accretion disks. Current laboratory experiments designed to reveal the instability in liquid metals will also be discussed.

Introduction

Gases or liquids containing mobile electrical charges are subject to the influence
of a magnetic field. In addition to hydrodynamical forces such as
pressure and gravity, an element of magnetized fluid also feels the Lorentz
force \(\boldsymbol J\times\boldsymbol B\ ,\) where \(\boldsymbol J\)
is the current density and \(\boldsymbol B\) is the magnetic field vector. If the fluid is in
a state of differential rotation about a fixed origin, this Lorentz force can
be surprisingly disruptive, even if the magnetic field is very weak. In particular,
if the angular velocity of rotation \(\Omega\) decreases with radial distance \(R\ ,\) the
motion is unstable: a fluid element undergoing a small displacement from circular
motion experiences a destabilizing force that increases at a rate which is itself
proportional to the displacement. This process is known as the Magnetorotational Instability, or "MRI".

The MRI is important in astrophysical settings, where differentially rotating
systems are very common and magnetic fields are ubiquitous. In
particular, thin disks of gas are often found around forming stars or in binary
star systems, where they are known as accretion disks. Accretion disks
are also commonly present in the centre of galaxies, and in some cases can
be extremely luminous: quasars, for example, are thought to originate from
a gaseous disk surrounding a very massive black hole. Our modern understanding
of the MRI arose from attempts to understand the behavior of
accretion disks in the presence of magnetic fields; it is now understood that
the MRI is likely to occur in a very wide variety of different systems.

The dynamics of what is now called the MRI were first studied in the
1950s by Chandrasekhar (Chandrasekhar 1953; 1961) and Velikhov (Velikhov, 1959). Although there was some follow-up
work in later decades (Fricke, 1969; Acheson and Hide
1972; Acheson and Gibbons 1978), the generality and power of the instability were not
fully appreciated until 1991, when Balbus & Hawley (Balbus and Hawley, 1991) gave a relatively simple
elucidation and physical explanation of this important process. Since that time, the MRI has
been extensively studied in many astrophysical environments.

Figure 1: The magnetorotational instability. Magnetic fields in a disk bind fluid elements precisely as though they were masses in orbit connected by a spring. The inner element mi orbits faster than the outer element mo, even though the former has less angular momentum. Thus, the spring causes a net transfer of angular momentum from mi to mo. This transfer is unstable, as described in the text. The inner mass continues to sink, whereas the outer mass rises farther outward. (Courtesy of H. Ji)

In a magnetized, perfectly conducting fluid, the magnetic forces behave in
some very important respects as though the elements of fluid were connected
with elastic bands: trying to displace such an element perpendicular to a
magnetic line of force causes an attractive force proportional to the displacement,
like a string under tension. Normally, such a force is restoring,
a strongly stabilizing influence that would allow a type of magnetic wave
to propagate. If the fluid medium is not stationary but rotating, however, attractive
forces can actually be destabilizing. The MRI is a consequence of this
surprising behavior.

Consider, for example, two masses, mi and mo connected by a spring under tension,
both masses in orbit abound a central body, Mc. In such a system, the angular
velocity of circular orbits near the center is higher than the angular velocity
of orbits farther from the center, but the angular momentum of the lower
orbits is smaller than that of the higher orbits. If mi is allowed to orbit a little
bit closer to the centre than mo, it will have a slightly higher
angular velocity. The connecting spring will pull back on mi, and drag mo
forward. This means that mi experiences a retarding torque, loses angular
momentum, and must fall to an orbit of smaller radius, corresponding to a smaller angular
momentum. mo, on the other hand, experiences a positive torque, acquires
more angular momentum, and moves to a higher orbit. The spring stretches
yet more, the torques become yet larger, and the motion is unstable! Because
magnetic forces act like a spring under tension connecting fluid elements, the
behavior of a magnetized fluid is almost exactly analogous to this simple
mechanical system. This is the essence of the MRI. (See Figure 1)

A more detailed explanation

To see this unstable behavior more quantitatively, consider the equations
of motion for a fluid element mass in circular motion with an angular
velocity \(\Omega\ .\) In general \(\Omega\) will be a function of the distance from the rotation axis \(R\ ,\) and we assume that the orbital radius is \(r=R_0\ .\) The centripetal force
required to keep the mass in orbit is \(-R\Omega^2(R)\ ,\) the minus sign
indicates a direction toward the center. If this force is gravity from
a point mass at the center, then the centripetal force is just \(-GM/R^2,\)
where \(G\) is Newton's constant and \(M\) is the central mass.

Let us now consider small departures from the circular motion of the
orbiting mass element caused by some perturbing force. We transform
variables into a rotating frame moving with the orbiting mass element at
angular velocity \(\Omega(R_0)=\Omega_0\ ,\) with origin located at the unperturbed,
orbiting location of the mass element. As usual when working in a
rotating frame, we need to add to the equations of motion a Coriolis
force \(-2\boldsymbol\Omega_0\times\boldsymbol v\) plus a centrifugal force \(R\Omega_0^2\ .\)
The velocity \(v\) is the velocity as measured in the rotating frame.
Furthermore, we restrict our attention to a small neighborhood near \(R_0\ ,\)
say \(R_0+x\ ,\) with \(x\) much smaller than \(R_0\ .\) Then the sum of the
centrifugal and centripetal forces is

to linear order in \(x\ .\) With our \(x\) axis pointing radial outward from
the unperturbed location of the fluid element and our \(y\) axis pointing in
the direction of increasing azimuthal angle (the direction of the unperturbed
orbit), the \(x\) and \(y\) equations of motion for a small departure from a circular
orbit \(R=R_0\) are:

where \(f_x\) and \(f_y\) are the forces per unit mass in the \(x\) and
\(y\) directions, and a dot indicates a time derivative (i.e.,
\(\dot x\) is the \(x\) velocity, \(\ddot x\) is the \(x\) acceleration, etc.).

In the absence of external forces, the equations of motion have solutions
with the time dependence \(e^{i\omega t}\ ,\) where the angular frequency \(\omega\)
satisfies the equation

where \(\kappa^2\) is known as the epicyclic frequency. In our solar
system, for example, deviations from a sun-centered circular orbit that
are familiar ellipses when viewed by an external viewer at rest, appear
instead as small radial and azimuthal oscillations of the orbiting element
when viewed by an observer moving with the undisturbed circular motion.
These oscillations trace out a small retrograde ellipse (i.e. rotating
in the opposite sense of the large circular orbit), centered on the
undisturbed orbital location of the mass element.

The epicyclic frequency may equivalently be written
\((1/R^3)(dR^4\Omega^2/dR)\ ,\) which shows that it is proportional to the
radial derivative of the angular momentum per unit mass, or specific
angular momentum. The specific angular momentum must increase outward
if stable epicyclic oscillations are to exist, otherwise displacements
would grow exponentially, corresponding to instability. This is a very
general result known as the Rayleigh criterion (Chandrasekhar 1961) for stability.
For orbits around a point mass, the specific angular momentum is
proportional to \(R^{1/2}\ ,\) so the Rayleigh criterion is well satisfied.

Consider next the solutions to the equations of motion if the mass element
is subjected to an external restoring force, \(f_x=-Kx\ ,\) \(f_y=-Ky\) where
\(K\) is an arbitrary constant (the "spring constant"). If we now seek
solutions for the modal displacements in \(x\) and \(y\) with time dependence
\(e^{i\omega t}\ ,\) we find a much more complex equation for \(\omega\ :\)

\[\tag{5}
\omega^4 - (2K+\kappa^2)\omega^2 +K(K+Rd\Omega^2/dR) =0\]

Even though the spring exerts an attractive force,
it may destabilize. For example,
if the spring constant \(K\) is sufficiently
weak, the dominant balance will be between the final
two terms on the left side of the equation. Then,
a decreasing outward angular velocity profile will produce negative values
for \(\omega^2\ ,\) and both positive and negative imaginary values for \(\omega\ .\)
The negative imaginary root results not in oscillations, but in exponential
growth of very small displacements. A weak spring therefore causes the
type of instability
described qualitatively at the end of the previous section. A strong
spring on the other hand, will produce oscillations, as one intuitively expects.

The spring-like nature of magnetic fields

To understand the how the MRI works, we must first understand the
conditions inside a perfectly conducting fluid in motion. This is
often a good approximation to astrophysical gases.
In the presence of a magnetic field \(\boldsymbol B\ ,\) a moving conductor
responds by trying to eliminate the Lorentz force on
the free charges. The magnetic force acts in such a way
as to locally rearrange these charges to produce an internal electric field
of \(\boldsymbol {E=-{v\times B}}\ .\) In this way, the direct Lorentz force
on the charges \(\boldsymbol {E+v\times B}\)
vanishes. (Alternatively, the electric
field in the local rest frame of the moving charges vanishes.)
This induced electric field can now itself induce further changes
in the magnetic field \(\boldsymbol B\) according to Faraday's law,

The equation of a magnetic field in a perfect conductor in motion
has a special property:
the combination of Faraday induction and zero Lorentz force
makes the field lines behave as though they were painted, or
"frozen," into the fluid. In particular, if \(\boldsymbol B\) is initially
nearly constant and \(\xi\) is a divergence-free displacement, then
our equation reduces to

so that \(\boldsymbol B\) changes only when there is a shearing displacement along the
field line.

To understand the MRI, it is sufficient to
consider the case in which \(\boldsymbol B\) is uniform in vertical \(z\) direction,
and \(\xi\) varies as \(e^{ikz}\ .\) Then

\[\tag{9}
\delta \boldsymbol B =ikB\boldsymbol \xi,\]

where it is understood that the real part of this equation expresses
its physical content. (If \(\boldsymbol \xi\) is proportional to \(\cos(kz)\ ,\)
for example, then \(\delta\boldsymbol B\) is proportional to \(-\sin(kz)\ .\))

A magnetic field exerts a force per unit volume on an electrically neutral,
conducting fluid equal to \(\boldsymbol J\times\boldsymbol B\ .\) With the help of the
Biot-Savart law \(\mu_0\boldsymbol {J=\nabla\times B}\ ,\) this becomes

The first term on the right is analogous to a pressure gradient. In our
problem it may be neglected because it exerts no force in the plane of the
disk, perpendicular to \(z\ .\) The second term acts like a magnetic tension
force, analogous to a taut string. For a small disturbance \(\delta\boldsymbol B\ ,\)
it exerts an acceleration given by

Thus, a magnetic tension force gives rise to a return force which is
directly proportional to the displacement. This means that the oscillation
frequency \(\omega\) for small displacements in the plane
of rotation of a disk with a uniform magnetic field in the
vertical direction satisfies an equation ("dispersion relation")
exactly analogous to equation (5),
with \(K={k^2B^2/\mu_0\rho}\ :\)
\[\tag{12}
\omega^4 + [2(k^2B^2/\mu_0\rho)+\kappa^2]\omega^2 +(k^2B^2/\mu_0\rho)
[(k^2B^2/\mu_0\rho)+Rd\Omega^2/dR] =0\]

As before, if \(d\Omega^2/dR<0\ ,\) there is an exponentially growing root of this
equation for wavenumbers \(k\) satisfying \((k^2B^2/\mu_0\rho)< - Rd\Omega^2/dR\ .\)
This corresponds to the MRI.

Notice that the magnetic field appears in equation (12)
only as the product \(kB\ .\) Thus, even if \(B\) is very small, for very
large wavenumbers \(k\) this magnetic tension can be important. This is
why the MRI is so sensitive to even very weak magnetic fields: their
effect is amplified by multiplication by \(k\ .\) Moreover, it can be shown
that MRI is present regardless of the magnetic field geometry, as long
as the field is not too strong.

In astrophysics, one is generally interested in the case for which the
disk is supported by rotation against the gravitational attraction of
a central mass. A balance between the Newtonian gravitational force
and the radial centripetal force immediately gives
\[\tag{13}
\Omega^2 = {GM\over R^3}\]

where \(G\) is the Newtonian gravitational constant,
\(M\) is the central mass, and \(R\) is radial location in the disk.
Since \(Rd\Omega^2/dR=-3\Omega^2<0\ ,\) this so called Keplerian disk
is unstable to the MRI. Without a weak magnetic field, the flow would
be stable.

For a Keplerian disk, the maximum growth rate is \(\gamma=3\Omega/4\ ,\) which
occurs at a wavenumber satisfying \((k^2B^2/\mu_0\rho)=15\Omega^2/16\ .\)
\(\gamma\) is very rapid, corresponding to an amplification factor of
more than 100 per rotation period.

The nonlinear development of the MRI into fully developed turbulence
may be followed via large scale numerical computation. Figure 2 is a
rendering of a black hole accretion disk in a turbulent state taken from a
recent calculation by J. Hawley. The colors represent different density
levels, with red being the largest and blue the smallest. Figure 3
is a local detail of a two-dimensional numerical study in which all flow
quantities are independent of azimuth. Here, angular momentum per unit
mass is plotted. Red colors indicate an angular velocity in excess of
the Keplerian value, while blue colors correspond to a deficit.

Figure 2: Density rendering from a three-dimensional numerical study of an accretion disk that is unstable to the magnetorotational instability. (Courtesy of J. Hawley)

Figure 3: A slice in a plane of constant azimuth taken from a small scale two-dimensional of the magnetorotational instability. Colors indicate specific angular momentum, with red corresponding to above the Keplerian value, and blue below it. (Courtesy of J. Hawley)

Applications and laboratory experiments

Interest in the MRI is based on the fact that it appears to give an explanation
for the origin of turbulent flow in astrophysical accretion disks (Balbus and Hawley, 1991).

A promising model for the compact, intense X-ray sources discovered in
the 1960s was that of a neutron star or black hole drawing in (“accreting”)
gas from its surroundings (Prendergast and Burbidge, 1968). Such gas always accretes with a finite amount of
angular momentum relative to the central object, and so it must first form a
rotating disk — it cannot accrete directly onto the object without first losing
its angular momentum. But how an element of gaseous fluid managed to
lose its angular momentum and spiral onto the central object was not at all
obvious.

One explanation involved shear-driven turbulence (Shakura and Sunyaev, 1973). There would be significant
shear in an accretion disk (gas closer to the centre rotates more rapidly
than outer disk regions), and shear layers often break down into turbulent
flow. The presence of shear-generated turbulence, in turn, produces the powerful
torques needed to transport angular momentum from one (inner) fluid
element to another (farther out).

The breakdown of shear layers into turbulence is routinely observed in
flows with velocity gradients, but without systematic rotation. This is an important
point, because rotation produces strongly stabilizing Coriolis forces,
and this is precisely what occurs in accretion disks. As can be seen in equation
(5), the K = 0 limit produces Coriolis-stabilized oscillations, not exponential
growth. These oscillations are present under much more general conditions
as well: a recent laboratory experiment (Ji et al., 2006) has shown stability of the flow profile
expected in accretion disks under conditions in which otherwise troublesome
dissipation effects are (by a standard measure know as the Reynolds number)
well below one part in a million. All of this changes, however, when
even a very weak magnetic field is present. The MRI produces torques that
are not stabilized by Coriolis forces. Large scale numerical simulations of
the MRI indicate that the rotational disk flow breaks down into turbulence (Hawley et al., 1995),
with strongly enhanced angular momentum transport properties. This is
just what is required for the accretion disk model to work. The formation of
stars (Stone et al., 2000), the production of X-rays in neutron star and black hole systems (Blaes, 2004), and
the creation of active galactic nuclei (Krolik, 1999) and gamma ray bursts (Wheeler, 2004) are all thought
to involve the development of the MRI at some level.

Thus far, we have focused rather exclusively on the dynamical breakdown
of laminar flow into turbulence triggered by a weak magnetic field,
but it is also the case that the resulting highly agitated flow can act
back on this same magnetic field. Embedded magnetic field lines are
stretched by the turbulent flow, and it is possible that systematic
field amplification could result. The process by which fluid motions
are converted to magnetic field energy is known as a dynamo (Moffatt, 1978); the two best
studied examples are the Earth's liquid outer core and the layers close
to the surface of the Sun. Dynamo activity in these regions
is thought to be responsible for maintaining the terrestrial and solar
magnetic fields. In both of these cases thermal convection is likely to
be the primary energy source, though in the case of the Sun differential
rotation may also play an important role. Whether the MRI is an efficient
dynamo process in accretion disks is currently an area of active research
(Fromang and Papaloizou, 2007).

There may also be applications of the MRI outside of the classical
accretion disk venue. Internal rotation in stars (Ogilvie, 2007), and even planetary
dynamos (Petitdemange et al., 2008) may, under some circumstances, be vulnerable to the MRI in
combination with convective instabilities. These studies are also
ongoing.

Finally, the MRI can, in principle, be studied in the laboratory (Ji et al., 2001), though
these experiments are very difficult to implement. A typical set-up
involves either concentric spherical shells or coaxial cylindrical shells.
Between (and confined by) the shells, there is a conducting liquid metal
such as sodium or gallium. The inner and outer shells are set in rotation
at different rates, and viscous torques compel the trapped liquid metal
to differentially rotate. The experiment then investigates whether the
differential rotation profile is stable or not in the presence of an
applied magnetic field.

A claimed detection of the MRI in a spherical shell experiment (Sisan et al., 2004), in
which the underlying state was itself turbulent, awaits confirmation
at the time of this writing (2009). A magnetic instability that bears
some similarity to the MRI can be excited if both vertical and azimuthal
magnetic fields are present in the undisturbed state (Hollerbach and Rüdiger, 2005). This is sometimes
referred to as the helical-MRI, (Liu et al., 2006) though its precise relation to the
MRI described above has yet to be fully elucidated. Because it is less
sensitive to stabilizing ohmic resistance than is the classical MRI,
this helical magnetic instability is easier to excite in the laboratory,
and there are indications that it may have been found (Stefani et al., 2006). The detection
of the classical MRI in a hydrodynamically quiescent background state
has yet to be achieved in the laboratory, however.